In the following the arguments from [1]} are adapted, which were previously applied to nonlinear Schrödinger equations. To avoid technicalities, we shall consider Schwartz initial data which admit global solutions and allow for integration by parts arguments. The a priori assumption can be removed later by well-posedness and limiting arguments.
We denote the linear part of the solution to (REF ) by
\(w(t) = \cos (t \sqrt{- \Delta }) f_1 + \frac{\sin (t \sqrt{- \Delta })}{\sqrt{- \Delta }} f_2.\) 
